Math Problem Statement

Indicate where f(x) = (x - 7) / ((x - 5)(x + 4)) is continuous.

Solution

The function provided in the image is:

f(x)=x7(x5)(x+4)f(x) = \frac{x-7}{(x-5)(x+4)}

To determine where f(x)f(x) is continuous, we need to identify the values of xx that cause the denominator to be zero since these will be the points of discontinuity.

The denominator is zero when:

(x5)(x+4)=0(x-5)(x+4) = 0

Solving this equation, we find:

x5=0orx+4=0x-5 = 0 \quad \text{or} \quad x+4 = 0 x=5orx=4x = 5 \quad \text{or} \quad x = -4

Therefore, the function f(x)f(x) is discontinuous at x=5x = 5 and x=4x = -4. Elsewhere, the function is continuous since the numerator and denominator are polynomials, and polynomials are continuous everywhere.

So, the function f(x)f(x) is continuous for all xx except at x=5x = 5 and x=4x = -4.

Would you like to explore how the limits behave at these points or have other questions on continuity or limits?

Here are five related questions you might consider:

  1. What is the behavior of f(x)f(x) as xx approaches 5?
  2. How does f(x)f(x) behave as xx approaches -4?
  3. Can f(x)f(x) be redefined at x=5x = 5 or x=4x = -4 to make it continuous at these points?
  4. What is the limit of f(x)f(x) as xx approaches infinity?
  5. How would the graph of f(x)f(x) look around the points of discontinuity?

Tip: When checking for continuity of a rational function, it’s helpful to factorize the denominator and look for common factors in the numerator, which can simplify the expression and potentially remove some points of discontinuity.

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Math Problem Analysis

Mathematical Concepts

Continuity
Rational Functions
Discontinuity

Formulas

f(x) = (x - 7) / ((x - 5)(x + 4))

Theorems

The Continuity Theorem for Rational Functions

Suitable Grade Level

Grades 11-12